Material or pecuniary damage is a reduction in or loss of assets, property damage, repair bills, medical bills, loss of wages, in some circumstances it could be a loss of profits - it has quantifiable monetary value.
Non-material or non-pecuniary damage is pain, fear, distress, anguish, anxiety, injury to feelings, affront to dignity, disfigurement, your suffering from the death of a loved one, even loss of enjoyment, or loss of future wages.
In the UK, note that before Google Inc v Vidal-Hall and Others (2015) a claim under s13 Data Protection Act 1998 for non-material damage could not be made without a claim for material damage. In that case the Court of Appeal held that a claim for non-material damage could be made without a claim for material damage and now as you point out GDPR makes it explicit that one can claim for material or non-material damage.
For example, from a leak of personal data, e.g. hundreds of thousands of customer records, someone might claim for material damage from actual identity fraud and/or claim for non-material damage from anxiety about the risk of becoming a victim of identity fraud resulting from the leak.